This invention relates to radiance modeling.
For the sake of simplicity, we limit the following background discussion to the modeling of up-welling radiances in a non-scattering terrestrial atmosphere bounded by a non-reflective surface and a single spectral observation (or channel). Let xcfx84v(z)represent the monochromatic transmittance at wavenumber v from top of the atmosphere (TOA) to an altitude z,                                                         τ              v                        ⁡                          (              z              )                                =                      ⅇ                          -                                                ∫                  z                  z0                                ⁢                                                      ∑                    m                                    ⁢                                      xe2x80x83                                    ⁢                                                                                    k                        v                        m                                            ⁡                                              (                        z                        )                                                              ⁢                                                                  ρ                        m                                            ⁡                                              (                        z                        )                                                              ⁢                                          xe2x80x83                                        ⁢                                          ⅆ                      z                                                                                                          ,                            Equation        ⁢                  xe2x80x83                ⁢        1            
where the discrete summation is over the different absorbing gases, kvm(z)is the pressure and temperature dependent absorption coefficient and xcfx81m(z)is the density of species m. The goal is to spectrally integrate the transmittances between TOA and a set of altitudes {zl;l=1, . . . , L} over the interval xcex94v of the measurement,                                           τ            ⁡                          (                              z                l                            )                                =                                    ∫                              Δ                ⁢                                  xe2x80x83                                ⁢                v                                      ⁢                                          φ                v                            ⁢                                                τ                  v                                ⁡                                  (                                      z                    l                                    )                                            ⁢                              xe2x80x83                            ⁢                              ⅆ                v                                                    ⁢                  xe2x80x83                ,                            Equation        ⁢                  xe2x80x83                ⁢        2            
where xcfx86v represents the normalized spectral response of a given instrument channel (optional) which is assumed to vanish outside of the interval.
For most practical applications, a direct numerical evaluation of the integral in Eq. (2) from monochromatic transmittances specified Dn a fine uniform spectral grid, such as those used by reference line-by-line models, is prohibitive.
Several approximate techniques have been proposed in the literature to perform this integration in a computationally efficient manner. The first class of techniques include the k-distribution (e.g., Lacis and Oinas, 1991; Goody et al., 1989) and Exponential Sum Fitting of Transmission functions (ESFT) (Wiscombe and Evans, 1977) methods, originally developed for a single absorbing species in a single atmospheric layer at given temperature and pressure. These methods approximate the transmittance of a single gas as a finite sum of weighted exponential functions,                               τ          ⁡                      (            u            )                          =                                            ∫              v                        ⁢                                          ⅇ                                                      -                                          k                      v                                                        ⁢                  u                                            ⁢                              xe2x80x83                            ⁢                              ⅆ                v                                              ≅                                    ∑                              i                =                1                            N                        ⁢                          xe2x80x83                        ⁢                                          w                i                            ⁢                                                ⅇ                                                            -                                              k                        i                                                              ⁢                    u                                                  .                                                                        Equation        ⁢                  xe2x80x83                ⁢        3            
In Eq.(4), u represents the path integrated absorber amount,
u=∫xcfx81z.xe2x80x83xe2x80x83Equation 4
The computational gain achieved with these approaches compared to the direct approach, originates from the non-monotonic nature of the molecular absorption spectrum and the fact that the value of the integral to the left of Eq. (3) is independent of the ordering of the k-values within the interval of interest. Elimination of the redundancy in the direct calculations is achieved by explicitly integrating over the probability distribution of k or by searching for a set of model parameters, {wi,ki}, that fits the transmittances evaluated for a range of values of u within a prescribed error threshold, as in the ESFT technique.
A number of approaches have been devised to apply the k-distribution and ESFT techniques to the treatment of non-homogenous atmospheres and gas mixtures (e.g. Lacis and Oinas, 1991; Goody et al., 1989; West et al., 1990; Armbruster and Fischer, 1996; Sun and Rikus, 1999). However, the use of simplifying assumptions limits the accuracy achievable with these proposed approaches and makes tuning and/or supervision a requirement for the model parameter generation. In addition, none of the proposed approaches addresses the full problem with the desirable characteristics that fitting errors monotonically decrease as the number of model parameters increase.
A second class of techniques for fast spectral integration of monochromatic transmittances includes the Frequency Sampling Method (FSM) or Radiance Sampling Method (RSM) (Tjemkes and Schmetz, 1997). Tjemkes and Schmetz attribute the limitations of the k-distribution method to the loss of wavenumber information during the generation of the model parameters and argue for the need to stay in wavenumber space. The FSM technique uses the following statistical estimator to approximate the integral of Eq. (2),                                                         τ              *                        ⁡                          (                              z                l                            )                                =                                    ∑                              i                =                1                            N                        ⁢                          xe2x80x83                        ⁢                                          φ                                  v                  i                                            ⁢                                                τ                                      v                    i                                                  ⁡                                  (                                      z                    l                                    )                                                                    ,                            Equation        ⁢                  xe2x80x83                ⁢        5            
where the vi""s are a set of arbitrarily selected wavenumbers within the interval spanned by the instrument function. For a given sampling strategy, the number, N, of samples used in the estimator controls the accuracy achieved with this method. This technique has the advantage that it is strictly monochromatic and that the effect of simultaneous absorption by the different gases and non-homogeneities in pressure and temperature are correctly accounted for. However, no criterion is used to objectively select the vi""s and the method does not fully eliminate the redundancies in the line-by-line calculations which results in an excessive number of samples used to achieve a required accuracy.
A final class of technique widely used in remote sensing applications (McMillin and Fleming, 1975, 1977, 1979; Eyre and Woolf, 1988; Eyre, 1991) parameterizes the xe2x80x9ceffective optical depthxe2x80x9d in each atmospheric layer as a function of atmospheric pressure, temperature and gas concentration,                                                         log              ⁡                              (                                                      τ                    ⁡                                          (                                              z                        l                                            )                                                                            τ                    ⁡                                          (                                              z                                                  l                          -                          1                                                                    )                                                                      )                                      -                          log              ⁡                              (                                                                            τ                      ref                                        ⁡                                          (                                              z                        l                                            )                                                                                                  τ                      ref                                        ⁡                                          (                                              z                                                  l                          -                          1                                                                    )                                                                      )                                              =                                    ∑                              k                =                1                            P                        ⁢                          xe2x80x83                        ⁢                                          a                k                            ⁢                              X                k                                                    ,                            Equation        ⁢                  xe2x80x83                ⁢        6            
where xcfx84ref(zl) represents the transmittance profile corresponding to a reference atmosphere, which can be computed exactly, and the Xk""s are functions of layer and path integrated (pressure weighted) temperature and gas concentrations. The weights ak are evaluated by least-square regression techniques to minimize the modeling error for an ensemble of globally representative atmospheric profiles (xe2x80x9ctrainingxe2x80x9d set). This technique has the advantage that it requires the evaluation of only a single exponential to compute the transmittance for each atmospheric level and molecular constituent. However, the accuracy of the method depends largely on the choice of empirical basis functions and is difficult to control. Moreover, different parameterizations may be required to fit the transmission of highly variable constituents such as water vapor in different absorption regimes.
The Optimal Spectral Sampling (OSS) method is a rapid and accurate physical technique for the numerical modeling of narrow band transmittances in media with non-homogeneous thermodynamic properties (i.e., temperature and pressure) containing a mixture of absorbing gases with variable concentrations.
In general, the invention features a method in which a spectrum is automatically searched to select individual spectral points, monochromatic calculations of transmittance are obtained at the selected spectral points, and a weighted sum of the monochromatic calculated transmittances is determined. The individual spectral points are selected so that the weighted sum is essentially exactly representative of calculations of average transmittances in spectral bands to which the selected spectral points belong.
Implementations of the invention may include one or more of the following features. The transmittances may be accurately and rapidly numerically modeled with respect to media containing a mixture of absorbing gases with variable concentrations and non-homogenous thermodynamic properties. The selected spectral points in modeling radiances measured by Earth-orbiting down-looking radiometers can be applied to any spectral domain and instrument viewing geometry, and to the general problem of flux or radiance computation in emitting and scattering atmospheres.
The spectral points are selected to minimize RMS differences between exact and estimated transmittance profiles (or radiances) for a predetermined set of atmospheric profiles.
The spectral points may be selected based on a set of uniformly spaced monochromatic transmittances obtained from reference line-by-line calculations. Each spectral point may be selected by an iterative process of finding a spectral location that produces a minimal error among M possible locations. Weights for the respective points may be re-evaluated in each iteration.
The method has been specifically designed for the modeling of radiances measured by Earth orbiting down-looking microwave and infrared radiometers but can be applied to any spectral domain and instrument viewing geometry, and to the general problem of flux or radiance computation in emitting and scattering atmospheres. The OSS method is particularly well suited for remote sensing applications and for the assimilation of satellite observations in Numerical Weather Prediction models. Both applications require fast and accurate models to produce radiances and radiance derivatives used for the real-time determination of profiles of atmospheric temperature and constituents from multi-spectral satellite observations (the so-called xe2x80x9cinversexe2x80x9d problem).
Other advantages and features will become apparent from the following description and from the claims.